Language is a biological trait that radically changed the performance of one species and the appearance of the planet. Understanding how human language came about is one of the most interesting tasks for evolutionary biology. Here we discuss how natural selection can guide the emergence of some basic features of human language, including arbitrary signs, words, syntactic communication and grammar. We show how natural selection can lead to the duality of patterning of human language: sequences of phonemes form words; sequences of words form sentences. Finally, we present a framework for the population dynamics of grammar acquisition, which allows us to study the cultural evolution of grammar and the biological evolution of universal grammar.